How to get formula for focus and directrix

Start by setting the distance from the point (x,y) to the focus (p,0) equal to the distance from the point (x,y) to the directrix x = -p. This will give you the equation (x-p)^2 + y^2 = (x+p)^2. From there, you can use algebra to rearrange the equation and get the standard form for a parabola, which is (x - p)^2 = 4p(y - 0). This will give you the formula for the focus (p,0) and the directrix x = -p. You can then follow a similar process for the focus (0,p) and directrix y = -p. This will give
  • #1
ilya.s
1
0
Hello everyone, this is an amazing forum! :)

I have a question: I have seen the formulas for focus/directrix of a parabola, but I cannot trust something unless I know how to get there.

How do I get the formulas of focus and directrix?

I got to an equation (X - Xf)^2 + (Y - Yf)^2 = (Y - Yd)^2 (distance between any point and focus equals to the distance between the same point and directrix). Where do I go from here?

Thank you!
 
Mathematics news on Phys.org
  • #2
You might start with a simple case where the focus and directrix are equidistant from the origin and dirextrix parallel to an axis: Focus (0,p) and directrix y = -p or symmetrically, Focus (p,0), directrix x = -p.

See http://en.wikipedia.org/wiki/Parabola.
 
  • #3
You can derive the formula using the distance formula.
 

Related to How to get formula for focus and directrix

1. What is the formula for finding the focus and directrix of a parabola?

The formula for finding the focus and directrix of a parabola is: Focus = (p/2, 0) and Directrix: y = -p/2, where p is the distance from the vertex to the focus or directrix.

2. How do you determine the value of p in the formula?

The value of p can be determined by the coefficient of the squared term in the equation of the parabola. If the equation is in the form y = ax^2 + bx + c, then p = 1/4a.

3. Can the focus and directrix be on the same side of the parabola?

No, the focus and directrix are always on opposite sides of the parabola. The focus is located inside the parabola while the directrix is a line outside the parabola.

4. How does changing the value of p affect the shape of the parabola?

The value of p determines the steepness of the parabola. A larger value of p will result in a narrower and steeper parabola, while a smaller value of p will result in a wider and flatter parabola.

5. Can the formula for focus and directrix be used for all parabolas?

Yes, the formula for focus and directrix can be used for all parabolas regardless of their orientation or position. It is a general formula that applies to all parabolas.

Similar threads

I Converse of focus-directrix property of conic sections
    • General Math
Replies
2
Views
469
B Parabola vs Hyperbola, why does a Hyperbola have two foci/curves?
    • General Math
Replies
3
Views
1K
MHB Find the equation of parabola with vertex (2, 1) and focus (1, -1)
    • General Math
Replies
4
Views
30K
MHB Find Equation of Parabola Given Focus & Directrix - Jose's Q&A
    • General Math
Replies
1
Views
2K
MHB Find Parabola Given Focus & Directrix - Help Needed
    • General Math
Replies
2
Views
5K
MHB Finding equation of parabola with focus and directrix
    • General Math
Replies
5
Views
13K
MHB Equation of ellipse from directrix and focus
    • General Math
Replies
1
Views
6K
Determining distance from Focus to Point on Parabola
    • General Math
Replies
1
Views
2K
Understanding Conic Sections: Exploring the Plane-Cone Intersection
    • General Math
Replies
1
Views
3K
Physical meaning of directrix?
    • General Math
Replies
1
Views
3K

Hot Threads

Recent Insights

Back
Top